For example, dinosaur ‘bones’, as I understand it, are fossils which are formed by sediment covering the organism quickly after death, forming a cavity which is later filled with minerals of some sort. And I take it there are other forms of fossilization as well.
But what are the oldest actual bones we have?
Replacement of bone by fossilized minerals is usually not complete. The bones of Sue the Tyrannosaur, for instance, were found to still contain soft marrow tissue, from which some of her DNA was extracted (just badly-degraded fragments, not nearly enough to go full Jurassic Park, but still definitely her actual DNA). I’d say that that’s enough to qualify those fossils as “actual bone”, though I don’t expect that’s the oldest such specimen.
There are impressions of very odd creatures in the Burgess Shale from British Columbia, about 500M years ago. Some are softbodied and some are apparently arthropods - though not (as I understand) bones. It’s unclear how much is impressions in the rock and how much is original material. I would assume a precise answer depends on how old animals with internal bones are, as opposed to external shell creatures.
Arthropod bones are very rare in the fossil record.
As for the OP’s question, I can’t find much (searches want to tell me about oldest human bones but the oldest unfossilized bones .ight be around 3 million years.
Thanks for the link. Interestingly, the first reply under that answer (at the time of this writing) was a link to a Hitler History Channel article that starts out,
“Unlike bones and teeth, which can survive for hundreds of millions of years, soft tissues are among the first materials to disappear during the fossilization process.”
Well, yeah, as arthropods dont have bones.
Rare as hen’s teeth they are.
Just as rare are vertebrates trapped in amber. This little critter - with unfossilised bone - is 110 million years old.
I was alluding to the hard shell - chitin? I assume that unlike bones or shells, which are calcium to start with, chitin is less likely to survive for eons. However, it does actually not decay the way flesh does, and is less soluable than calcium. A quick Google suggests it is still detectable in some fossils.
As of this morning, the oldest bones are roughly 5,000,064 years old; that was me getting out of bed. 
I’d like to read this paper if you have it (or better search terms — mine came up dry.)
My guess would be that the earliest bones that existed were likely from the point where cartilaginous fishes started developing sturdier skeletons. The oldest bony fish fossil is about 410 million years old (according to the Imperial College London website), and ‘life moved from the sea to the land’ about 390 million years ago.
For non-fossilized bones, there are 42,000-year-old frozen mummified animals being found in the Arctic.
Here’s an article from Discover magazine back in 2006. Don’t know if that’s what @Chronos was referring to?
I thought so, too, but an earlier SDMB thread steered me straight. It turns out that hard bones actually evolved before cartilage.
The article from 2020 linked below talks about the ‘recent’ discovery of a 410-million-year-old bony fish skull. It mentions sharks with cartilaginous skeletons evolved from fishes that were bony, but those had in turn evolved from earlier cartilaginous fishes. It sounds like they keep revising their theories as they find more fossils. (As they should.)
Meant to include this quotation from the article I linked:
Once, when she was working with a T. rex skeleton harvested from Hell Creek, she noticed that the fossil exuded a distinctly organic odor. “It smelled just like one of the cadavers we had in the lab who had been treated with chemotherapy before he died,” she says. Given the conventional wisdom that such fossils were made up entirely of minerals, Schweitzer was anxious when mentioning this to Horner. “But he said, ‘Oh, yeah, all Hell Creek bones smell,’” she says. To most old-line paleontologists, the smell of death didn’t even register. To Schweitzer, it meant that traces of life might still cling to those bones. She had already seen signs of exceptional preservation in the early 1990s, while she was studying the technical aspects of adhering fossil slices to microscope slides. One day a collaborator brought a T. rex slide to a conference and showed it to a pathologist, who examined it under a microscope. "The guy looked at it and said, ‘Do you realize you’ve got red blood cells in that bone?’ " Schweitzer remembers. “My colleague brought it back and showed me, and I just got goose bumps, because everyone knows these things don’t last for 65 million years.”
whatever the numbers are in the article, make sure to add 18 years … 
Is this accurate? I thought that the organic materials were gradually replaced by the fossilization process, not that the fossils are a cast?
I believe you are both right, but you are righter than him (from Wikipedia):
There are many processes that lead to fossilization, including permineralization, casts and molds, authigenic mineralization, replacement and recrystallization, adpression, carbonization, and bioimmuration.
That article lists several methods, why is one more right?
I understand it is more frequent. There are many biases here (more frequently found, more frequently exposed in a museum, more frequently reviewed in an article…).